How do I take the integral of complex power function with a non-integral power?

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I cannot find any references talking about how to do this, and I cannot figure it out. Suppose we have

$\int_\Gamma z^{a/b} dz$,

where $z \in \mathbb{C}$ and $\Gamma$ is some contour which has an endpoint on the negative real axis.

I know that we need to convert $z^{a/b}$ to $e^{a/b \ \text{Log}(z)}$, but then I don't know what to do. Any help would be appreciated.

(I am aware there is a question asked with an almost identical title, but it is not helpful and has no answer.)